A perturbative approach to the polaron shift of excitons in transition
metal dichalcogeniedes
J. C. G. Henriques\(^{1,2}\)  and N. M. R. Peres\(^{1,2}\) 
\(^{1}\) Department and Centre of Physics, and QuantaLab, University
of Minho, Campus of Gualtar, 4710-057, Braga, Portugal
\(^{2}\) International Iberian Nanotechnology Laboratory (INL), Av. Mestre
JosÃ© Veiga, 4715-330, Braga, Portugal

In this paper we study the phonon's effect on the position of the
1s excitonic resonance of the fundamental absorption transition line
in two-dimensional transition metal dichalcogenides. We apply our
theory to WS\(_{2}\) a two-dimensional material where the shift in absorption
peak position has been measured as a function of temperature. The
theory is composed of two ingredients only: i) the effect of longitudinal
optical phonons on the absorption peak position, which we describe
with second order perturbation theory; ii) the effect of phonons on
the value of the single particle energy gap, which we describe with
the Huang Rhys model. Our results show an excellent agreement with
the experimentally measured shift of the absorption peak with the
temperature.

One of the most prominent and studied types of two-dimensional materials
are transition metal dichalcogeniedes (TMDs) [1]}.
These, often semi-conducting, materials present remarkable electronic
and optical properties, intrinsically related with their excitonic
response. An exciton is a quasi-particle corresponding to a bound
electron-hole pair interacting via a Coulomb-like potential. Due to
the reduced dielectric screening in two-dimensional materials, these
quasi-particles are more tightly bound, and thus more stable, than
their three-dimensional analogues. Phonons, the quanta of atomic vibrational
energy, are known to have a significant impact on the optical properties
of TMDs, especially due to their interaction with excitons [2]}, [3]}, [4]}, [5]}.
The exciton-phonon coupling influences both the line width and the
peak position of the different absorption resonances associated with
the optically active excitonic states in TMDs. Indeed, this effect
has been reported recently in Ref.[6]}, where
it was shown that the \(1s\)  excitonic peak was red-shifted as the
temperature increased, accompanied by a concomitant increased line
width of the resonance. The coupling of phonons to excitons also affects
their radiative lifetime, and allows the access to optically dark
states via inter-valley scattering channels [7]}.
Studies on the temperature dependence of the optical properties of
these materials are highly relevant to accurately predict their applicability
in different technological applications which are required to work
at room temperature.
